The rule that only sheep and ram wool are forbidden is implied in the first Mishna in the chapter that deals with Shaatnez (Kilayim 9:1) and is derived in the Yerushalmi there from the verse in II Kings 3:4 which describes ram and sheep wool with the sole word "wool"; all other types of wool which have שם לווי = an associated adjective (such as goat wool) are not included.

Obviously, if the cashmere garment has extra threads or fibers used for stitching or padding then there could still be an issue of Shaatnez, but at least there isn't any more likelihood qua cashmere than other neutral materials a priori.

Although the other answers mentioned that cashmere may be mixed with linen as it comes from the Kashmir goat, practically this is not true. That is because cashmere by itself is a very expensive material. Each goat can only produce about 4 ounces of fibers a year and it is estimated that fleece from 4 to 6 animals would be needed for a sweater. Because of the expense, cashmere is all most always found mixed with regular sheep's wool. Therefore, mixing cashmere with linen would be a problem of shatnez.

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msh210♦Dec 31 '12 at 15:20

"There is no need to check for Shaatnez on any article of clothing unless you're sure there's shaatnez in there.

For all those who need to know....
Shaatnez according to some Rishonim requires "Shua, Tavi, and Nuz all together." We don't have that today. So according to those Rishonim, today's Shaatnez is only derabanan. When you bring a suit home, you only have a Safek Derabanan that is not "efsher levrury bekal." Or you can see it as a Sefek sefeka on a de'oraita. Same difference.
The current so-called Shaatnez awareness was not initiated by Talmidei Chachamim."